Here are examples I was able to change. After that are examples I have trouble with.

He heard sirens approaching in the distance. << I noticed that in the distance is redundant.
Approaching sirens whined.

He knew he should ignore these people and run.
He should ignore these people and run.

He was aware of the people watching him.
The surrounding people watched him.

He felt her tail wrap around his lower back.
Her tail wrapped around his lower back.

The man fired. Bharita saw the dart exit the gun as if in slow motion.
The man fired, and the dart exited the gun as if in slow motion.

For a moment he felt his sanity slipping.
For a moment his sanity slipped.

It seems most of the time you can remove these words by changing the simple past tense.

Sometimes I have trouble thinking of ways to remove said words.

Sam pressed her back against Bharita’s chest. He could feel her heart thumping.

How should I remove felt?

Turn both sentences into this? She pressed back into Bharita, and her heart thumped against his chest.

On the other side of that door, Bharita saw officers sitting at desks and processing paper work.

Bharita has the ability to see by sensing the frequencies of all objects, similar to how a bat uses sonar. If this fact is already well established, can I write? On the other side of that door officers sat at desks, processing paper work.

Normally that would be telling information a character could not know, but in this case Bharita does know.

It's niot necessary to remove every instance. For instance, you might want to leave "he noticed" in place to emphasize the moment of realization of some event. But if you do it repeatedly, you're pulling the reader out of te character's head and turning the reader from participant to observer.

As for the "best" way, that always gives me pause. It depends on what you want to emphasize. Making her heart the subject, thumping against his cheek, emphasizes te sensation while downplaying the character. It's more passive than making him the subhect of tye sentence, but still is probably a more powerfuls sentence in context of a tender moment.

Most of the time, you won;t want to evaluate a crirical passage in terms of a single guideline. Instead, you'll look for the best verb for a sentence, and then see if that turns the focus of te sentence away from what you wanted, or lets the POV slip, etc. etc.

But if you find the the sensory indirection is getting in the way, the repair strategy above is a good one. Just be sensible.